Sophie, I'm not wanting to be flippant, really. But we do have many words that share the same root. "Trans" means to cross over and it makes sense that many of the words in this community would share that root.
People confuse the two because they are ignorant of the fact that gender is separate from sexuality. I mean, most people are gender conformant and they are hetero. They take it that gender and sexuality are tied together, it is "normal" to be opposite-sex attracted, they don't know the difference between a person who wants to change their sexual characteristics and someone who doesn't but wants to express femininity, and they don't immediately think that "sex", "sexual", or "sexuality" means both, a person's biology and also their attraction to others. They don't take the time to think deeply about gender and sexuality issues. This is why we seriously need to teach our kids what this is all about.
